Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.17032 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915126467100672 |
|---|---|
| author | Glogić, Irfan Hofmanová, Martina Lange, Theresa Luongo, Eliseo |
| author_facet | Glogić, Irfan Hofmanová, Martina Lange, Theresa Luongo, Eliseo |
| contents | We consider the focusing power nonlinearity heat equation
\begin{equation}\label{Eq:Heat_abstract}\tag{NLH}
\partial_t u -Δu = |u|^{p-1}u, \quad p>1,
\end{equation} in dimensions $d \geq 3$. It is well-known that if $p$ is large enough then \eqref{Eq:Heat_abstract} is unconditionally locally well-posed in $L^q(\mathbb{R}^d)$ for $q \geq d(p-1)/2$. We prove that this result is optimal in the sense that uniqueness of local solutions fails when $q < d(p-1)/2$ as long as $p < p_{JL}$, where $p_{JL}$ stands for the Joseph-Lundgren exponent. Our proof is based on the method that Jia-Šverák proposed in \cite{JiaSve15} to show non-uniqueness of Leray solutions to incompressible 3d Navier-Stokes equations. In particular, we rigorously verify for \eqref{Eq:Heat_abstract} the (analogue of the) spectral assumption made in \cite{JiaSve15}. To our knowledge, this is the first rigorous implementation of the Jia-Šverák method to a nonlinear parabolic equation without forcing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17032 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of mild solutions to supercritical heat equations Glogić, Irfan Hofmanová, Martina Lange, Theresa Luongo, Eliseo Analysis of PDEs Spectral Theory We consider the focusing power nonlinearity heat equation \begin{equation}\label{Eq:Heat_abstract}\tag{NLH} \partial_t u -Δu = |u|^{p-1}u, \quad p>1, \end{equation} in dimensions $d \geq 3$. It is well-known that if $p$ is large enough then \eqref{Eq:Heat_abstract} is unconditionally locally well-posed in $L^q(\mathbb{R}^d)$ for $q \geq d(p-1)/2$. We prove that this result is optimal in the sense that uniqueness of local solutions fails when $q < d(p-1)/2$ as long as $p < p_{JL}$, where $p_{JL}$ stands for the Joseph-Lundgren exponent. Our proof is based on the method that Jia-Šverák proposed in \cite{JiaSve15} to show non-uniqueness of Leray solutions to incompressible 3d Navier-Stokes equations. In particular, we rigorously verify for \eqref{Eq:Heat_abstract} the (analogue of the) spectral assumption made in \cite{JiaSve15}. To our knowledge, this is the first rigorous implementation of the Jia-Šverák method to a nonlinear parabolic equation without forcing. |
| title | Non-uniqueness of mild solutions to supercritical heat equations |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2501.17032 |